Normal-mode-based analysis of electron plasma waves with second-order Hermitian formalism

TL;DR

This paper presents a second-order Hermitian formalism for analyzing Langmuir electron waves in collisionless plasma, providing a complete basis of normal modes and a clear expression for their evolution, including Landau damping effects.

Contribution

It introduces a novel second-order self-adjoint framework for plasma wave analysis, enabling explicit mode decomposition and solution for initial-value problems.

Findings

Complete basis of singular normal modes derived

Orthogonality relations established for modes

Explicit solution illustrating Landau damping

Abstract

The classic problem of the dynamic evolution of Langmuir electron waves in a collisionless plasma and their Landau damping is cast as a second-order, self-adjoint problem with a continuum spectrum of real and positive squared frequencies. The corresponding complete basis of singular normal modes is obtained, along with their orthogonality relation. This yields easily the general expression of the time-reversal-invariant solution for any initial-value problem. An example is given for a specific initial condition that illustrates the Landau damping of the macroscopic moments of the perturbation.